On the representation dimension of smash products

Let \(A\) be a finite dimensional \(G\)-graded algebra with \(G\) a finite group, and \(A\# k[G]^{\ast}\) be the smash product of \(A\) with the group \(G\). Our results can be stated as follows: (1) If \(A\) is a self-injective algebra and separably graded, then the dimensions of triangulated categories \(\underline{\rm mod}A\) and \(\underline{\rm mod}A\# k[G]^{\ast}\) are equal. In particular, we obtain that the representation dimension of \(A\# k[G]^{\ast}\) is at least the dimension of triangulated category \(\underline{\rm mod}A\) plus 2; (2) Generally, if \(A\) is a \(k\)-algebra and separably graded, then the Oppermann dimensions of \(A\) and \(A\# k[G]^{\ast}\) are equal. In particular, we obtain that the representation dimension of \(A\# k[G]^{\ast}\) is at least the Oppermann dimension of \(A\) plus 2. In the end, we give two examples to illustrate our results.